Rectangle
From Wikipedia, the free encyclopedia
| Rectangle | |
|---|---|
 | |
| Type | Quadrilateral |
| Edges and vertices | 4 |
| Schläfli symbol | {}x{} |
| Symmetry group | D2 (*2) |
| Coxeter-Dynkin diagram |   |
| Dual polygon | Rhombus |
| Properties | isogonal, convex, cyclic |
In Euclidean geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle.
A rectangle that is not simple is complex, but more clearly described as self-intersecting or crossed. It is defined as a self-intersecting quadrilateral with the same vertex arrangement as a simple rectangle.
Rectangles may be used in periodic tilings of the plane. Another popular subject in recreational mathematics is the tiling of rectangles by polygons, ranging from simple puzzles to unsolved problems.
Contents |
[edit] General properties
Every rectangle, simple or crossed, has the following properties:
- It is isogonal.
- The two diagonals are equal in length.
- Opposite sides are equal in length.
- It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
- The dual polygon of a rectangle is a rhombus.
[edit] Simple rectangle
A simple rectangle has the following properties:
- It is convex and cyclic.
- All angles are 90 degrees.
- The two diagonals bisect each other.
- Opposite sides are parallel.
A simple rectangle is a special case of a parallelogram, which has two pairs of parallel opposite sides. A parallelogram, and hence also a rectangle, is a special case of a trapezium (known as a trapezoid in North America), which has at least one pair of parallel opposite sides.
If a simple rectangle has length l and width w
- it has area A = lw,
- it has perimeter P = 2l + 2w = 2(l + w),
- each diagonal has length
, - and when l = w, the rectangle is a square.
The term oblong is occasionally used to refer to a non-square simple rectangle.[1][2]
Two simple rectangles, neither of which will fit inside the other, are said to be incomparable.
[edit] Crossed rectangle
A crossed rectangle is a complex (self-intersecting) rectangle, also called a bow-tie rectangle or butterfly rectangle.
It has the same vertex arrangement as a simple rectangle with which it shares two edges. Its other two edges are the diagonals of the simple rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.
The interior of a crossed rectangle can have a polygon density of +/-1 in each half triangle, dependent upon the winding orientation as clockwise or counterclockwise. To make a text rectangle hold down alt and press 2267 while alt is held down and you will get this image : █
[edit] Tessellations
The rectangle is used in many periodic tessellation patterns, in brickwork, for example, these isogonal tilings:
 Stacked bond |  Running bond |  Basket weave |  Basket weave |  Herringbone pattern |
[edit] Squared, perfect, and other tiled rectangles
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangled" (or "triangulated") rectangle respectively. The tiled rectangle is perfect[3][4] if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is imperfect. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles.
A rectangle has commensurable sides if and only if it is tilable by a finite number of unequal squares.[3][5] The same is true if the tiles are unequal isosceles right triangles.
The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.
[edit] See also
[edit] References
- ^ http://www.mathsisfun.com/definitions/oblong.html
- ^ http://www.icoachmath.com/SiteMap/Oblong.html
- ^ a b R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte (1940). "The dissection of rectangles into squares". Duke Math. J. 7 (1): 312–340. doi:10.1215/S0012-7094-40-00718-9. http://projecteuclid.org/euclid.dmj/1077492259.
- ^ J.D. Skinner II, C.A.B. Smith and W.T. Tutte (November 2000). "On the Dissection of Rectangles into Right-Angled Isosceles Triangles". J. Combinatorial Theory Series B 80 (2): 277–319. doi:10.1006/jctb.2000.1987.
- ^ R. Sprague (1940). "Ũber die Zerlegung von Rechtecken in lauter verschiedene Quadrate". J. fũr die reine und angewandte Mathematik 182: 60–64.
[edit] External links
 | Wikimedia Commons has media related to category: Rectangles |
- Weisstein, Eric W., "Rectangle" from MathWorld.
- Definition and properties of a rectangle With interactive animation
- Area of a rectangle with interactive animation
